Problem: How many integers satisfy $-4 \le 2x+1 \le 6$?
Subtracting $1$ from each expression, we have $-5\le 2x\le 5$. Dividing by $2$, we get $-\frac52\le x\le \frac52$. The integers satisfying this chain of inequalities are $-2,-1,0,1,2$. There are $\boxed{5}$ numbers in this list.